Analytical computation of boundary integrals for the Helmholtz equation in three dimensions

TL;DR

This paper derives analytical formulas for boundary integrals involving the Helmholtz Green's function over flat polygons, enabling more efficient and accurate computation in boundary element methods.

Contribution

It introduces explicit analytical expressions for boundary integrals of the Helmholtz equation over polygons, reducing reliance on numerical quadrature.

Findings

Provides convergent infinite series representations for boundary integrals.

Enables efficient truncation with error bounds for practical computation.

Facilitates integration into fast algorithms like the Fast Multipole Method.

Abstract

A key issue in the solution of partial differential equations via integral equation methods is the evaluation of possibly singular integrals involving the Green's function and its derivatives multiplied by simple functions over discretized representations of the boundary. For the Helmholtz equation, while many authors use numerical quadrature to evaluate these boundary integrals, we present analytical expressions for such integrals over flat polygons in the form of infinite series. These can be efficiently truncated based on the accurate error bounds, which is key to their integration in methods such as the Fast Multipole Method.